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Duality is the operation that interchanges hypervertices and hyperfaces onoriented hypermaps. The duality index measures how far a hypermap is from beingself-dual. We say that an oriented regular hypermap has \emph{duality-type}$\{l,n\}$ if $l$ is the valency of its vertices and $n$ is the valency of itsfaces. Here, we study some properties of this duality index in oriented regularhypermaps and we prove that for each pair $n$, $l \in \mathbb{N}$, with $n,l\geq 2$, it is possible to find an oriented regular hypermap with extremeduality index and of duality-type $\{l,n \}$, even if we are restricted tohypermaps with alternating or symmetric monodromy group.

Duality on hypermaps with symmetric or alternating monodromy group